Chapter 4 - Vector algebra Flashcards
What are direction cosines?
The direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three coordinate axes.
Define OP’s direction cosines (l, m, n) in regards to x, y and z.
l = x / r
m = y / r
n = z / r
If P har coordinates (2, -1, 3), find the length oP and the direction cosines of OP.
OP2 = (2)2 + (-1)2 + (3)2
= 4 + 1 + 9
This gives OP = sqrt(14)
Direction cosines are:
l = 2*sqrt(1/14)
m = -sqrt(1/14)
n = 3sqrt(1/14)
When are two vectors equal?
Two vectors a and b are equal if and only if they have the same modulus and the same direction and sense.
In component form, two vectors are equal if and only if the components are equal, that is:
a1 = b1, a2 = b2, a3 = b3
If k is a scalar and the vectors are related by a = kb then:
if k > 0, a is a vector in the ____ as b with magnitude k times the magnitude of b.
same direction
If k is a scalar and the vectors are related by a = kb then:
if k < 0, a is a vector in the ____ as b with magnitude k times the magnitude of b.
opposite direction
What is a vectors modulus?
A vectors modulus is a vectors length or magnitude. It’s written as |a| or |OA| (with arrow on top).
What is a unit vector?
A unit vector is a vector with modulus 1.
It’s written with a hat: â
â = a / |a|
What does the triangle law say about vector addition?
If two vectors a and b are represented in magnitude and direction by the two sides of a triangle taken in order then their sum is represented in magnitude and direction by the closing third side.
What does the commutative law say?
a + b = b + a
Order does not matter
What does the associative law say?
The brackets do not matter and can be omitted.
(a+b) + c = a + (b + c)
What does the distributive law say?
k(a + b) = ka + kb
We can multiply brackets out by the usual laws of algebra.
How do you subtract vectors?
We define subtraction in the obvious way:
a - b = a + (-b)
Where -b is the same vector as b, but facing the opposite way.
What letters are most commonly used to describe the three coordinate unit vectors in 3D-space?
i, j and k.
i for x-axis, j for y-axis, and k for z-axis.
How do you add the vectors (a1, a2, a3) and b1, b2, b3)?
By adding each of the respective components together:
(a1+b1, a2+b2, a3+b3)
How do you multiply a vector with a scalar? (Component form)
If a = (a1, a2, a3), then ka is:
(ka1, ka2, ka3)
How is the scalar (or dot or inner) product of two vectors a = (a1, a2, a3) and b = (b1, b2, b3) defined? (Both component and geometrical form)
In components:
a • b = a1b1 + a2b2 + a3b3 (= a scalar number)
a • b = | a | | b | cos ß
Consider that the scalar product of two vectors is equal to zero. What does this imply?
The two vectors are perpendicular.
a • b = 0
can mean three things. What are they?
either a = 0,
or b = 0,
or the angle between a and b is 90 degrees (they are perpendicular).
Does a • b = a • c
mean that b = c?
Not necessarily. There are three possible solutions:
Either, b = c
or a = 0
or a is perpendicular to b - c.
Consider the unit vectors i, j and k.
What is i • j
equal to?
i • j = j • k = k • i = 0
because the unit vectors are mutually perpendicular.
Given the two vectors a and b, what is the geometrical definition of the vector product?
a x b = | a | | b | sin ß ñ,
where ß is the angle between a and b (and between 0 and π), and ñ is the unit vector perpendicular to both a and b, such that a, b, ñ form a right-handed set.
The vector product of two vectors is itself a vector.
What does the anti-commutative law for vector products say?
a x b = -(b x a)
The distributive law over addition for vector products states that
(a x b) + (a x c)
can be written as ___
a x (b + c)
Given the two vectors a = (a1, a2, a3) and b = (b1, b2, b3), how can the vector product be defined in component form?
a x b can be defined in determinant form (actually an accepted misuse of the determinant form)
The vector product of two vectors is also the area of ____
The parallelogram defined by the two vectors (++ (du forstår!))
Where the vector product defines the area of the parallelogram, the triple scalar product defines the _____
Area of the parallelepiped
In cartesian/component form, the triple scalar product can be written as:
The determinant
Describe in a few steps how to find the area of a triangle using vector product, given the three vertices A, B and C of the triangle.
- Find two vectors starting from the same point, e.g. AB and AC.
- Find the vector product (AB x AC).
- Since the area of the parallelogram is the same as the length of the vector product, the area of the triangle is half of that.
Area of triangle = 1/2 * |(AB x AC)|
Describe in a few steps how to find the constant k such that the three vectors (7, 3, -1), (1, -1, 7) and (2, -4, k) are coplanar.
(You don’t need to do the calculations)
- If the three vectors are coplanar, they have a common perpendicular vector. Since we have two whole vectors, we can find the vector product of those. The vector product is itself a vector, and is perpendicular to the two vectors.
- We now have the perpendicular vector (a). We want a • (2, -4, k) = 0
to be true. Therefore, we manipulate the scalar product to get k alone.
The answer to this particular problem is 24.
Describe in a few steps how to solve this problem:
If a and b are perpendicular, simplify (a - 3b) • (8a + 5b)
- When multiplying for scalar product like this, you can solve the expression like a regular algebraic expression. This particular problem will result in (8a2-19ab-15b2).
- Since we know that a and b are perpendicular, (19ab) will result in 0, since ab = 0.
- Thus, the simplified expression is 8a2-15b2.
Describe in a few steps how to solve this problem:
Find the angle between p = 8i - 5j and q = -11i + 9j
- We know that |p| |q| cosß = 8*(-11) + (-5)*9.
The left side is the geometrical definition of the scalar product, while the rigt side is the definition of the scalar product on component form.
- From there, we manipulate the equation to get cosß alone on one side of the equation.
